Analytic geometry

02182019, 05:51 PM
(This post was last modified: 02202019 02:18 PM by Albert Chan.)
Post: #4




RE: Analytic geometry
Since we know x cos(m) + y sin(m) = 1 is just tangents to the unit circle, we can simplify:
Rotate the angles, so that t1 = 0, line 1 is now simply x*1 + y*0 = x = 1: Intersect, x=1 and line 2: 1*cos(t2) + y*sin(t2) = 1 (1  2 sin(t2/2)^2) + y*sin(t2) = 1 y * (2 sin(t2/2) cos(t2/2)) = 2 sin(t2/2)^2 y = tan(t2/2) Use symmetry, intersect of x=1 and line 3: y = tan(t3/2) > Triangle line 1 side length =  tan((t2t1)/2)  tan((t3t1)/2)  (02182019 09:27 AM)yangyongkang Wrote: Randomly take three parameters and draw this image Without solving for the vertice coordinates, we can get side length directly. Example, for above plot: +1 radian line, slope ≈ 0.642: length =  tan((81)/2  tan((31)/2)  ≈ 6.1947 −8 radian line, slope ≈ 0.147: length =  tan((1+8)/2  tan((3+8)/2)  ≈ 5.6329 +3 radian line, slope ≈ 7.015 : length =  tan((13)/2  tan((83)/2)  ≈ 2.5530 Edit: you can solve the vertices thru "reverse" rotation Example, the top vertice (line 1 and line 3 intersect): We know rotated intersect was [1, tan((31)/2)], so original intersect = [[cos(1), sin(1)], [sin(1), cos(1)]] * [1, tan(1)] = [cos(1)  sin(1)*tan(1), sin(1) + cos(1)*tan(1)] = [cos(2)/cos(1), 2*sin(1)] ≈ [0.7702, 1.6829] 

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Messages In This Thread 
Analytic geometry  yangyongkang  02182019, 09:27 AM
RE: Analytic geometry  yangyongkang  02182019, 12:09 PM
RE: Analytic geometry  Albert Chan  02182019, 08:48 PM
RE: Analytic geometry  Albert Chan  02182019, 03:16 PM
RE: Analytic geometry  Albert Chan  02182019 05:51 PM
RE: Analytic geometry  Albert Chan  02192019, 10:23 PM
RE: Analytic geometry  Albert Chan  02202019, 09:33 PM

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